Abstract
The financial crises that have occurred since 2006 have been associated with a degradation in financial ethics. Since the teaching of derivative pricing is often undertaken in the context of abstract mathematics, the question arises of the role of mathematics in supporting financial ethics. At the heart of the Fundamental Theorem of Asset Pricing, the foundational theory of mathematical approaches to derivative pricing, is the concept of reciprocity. This chapter shows how reciprocity was reliant upon the emergence of probability theory before 1700, where a risk-less profit was seen as illicit. The chapter finishes with a discussion of how this ethical approach to financial economics is presented to undergraduate and, more advanced, post-graduate students.
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Notes
- 1.
Blaise Pascal (1623–1662) was a sickly boy but very bright, and when he was eight, his father, Etienne, a judge, moved his family to Paris in order to focus on his son’s education. In Paris, both Blaise and Etienne became involved in a circle of mathematicians centred on the Abbé Mersenne. In this environment, Pascal wrote his first mathematical text, when he was 16, which was so clear that some thought it brilliant, while others, including Descartes , thought it was written by the father. However, when he was about 18, Pascal’s poor constitution caught up with his brilliant mind and he had a physical breakdown and had to give up working for around four years. Sometime after his recovery the Pascal family became Jansenists, fundamental Augustinians within the Catholic Church—‘Calvinist Catholics’—after Etienne had been treated at the Abbey of Port-Royal in Paris, which was a centre of the sect. In 1651 Etienne Pascal died, leaving a small fortune to his only son, and the 28-year-old Blaise Pascal was an independent man with a private income . Pascal abandoned the austere aspects of Jansenism and became involved with a group around the Duke of Roannez which included a noted gambler, the Chevalier de Méré who probably introduced Pascal to the Problem of Points.
In October 1654, two months after he had solved the Problem of Points, Pascal was involved in an accident and narrowly escaped death. Whether or not Pascal took this as a divine sign is disputed, but late the following November Pascal had an intense religious experience, retired to the centre of Jansenism, the convent of Port-Royal, and stopped working on mathematics. Pascal directed his efforts to supporting the Jansenist cause, primarily in his Lettres Provinciales (Provincial Letters) which was a popular satirical attack on casuistry, the idea that an act, such as lying, might be ethically acceptable under particular circumstances, which was popular amongst the Jesuits.
After his death, Pascal’s most influential philosophical work was published, his Pensèes (‘Thoughts’). These were an incoherent collection of ideas that Pascal, not having recovered from his various breakdowns. In the middle of these ideas, on two pieces of paper in a mess of writing are the details of Pascal’s Wager, his argument that people should believe in God.
- 2.
Pierre Fermat (1607–1665) was born in Gascony in south-west France where his father was a rich leather merchant who had a position in local government and sent Pierre to study law at the University of Toulouse and then Orleans. And at the age of 30, Fermat was admitted as a counsellor, or lawyer, at the regional court, the Parlement of Toulouse, and was appointed a King’s Counsellor in 1648. Fermat was known for his wisdom—he was literate in Greek, Latin, Spanish as well as French—and his good temper and kindness. Fermat did not make his living out of mathematics and this meant that he rarely published proofs, what we know about his mathematics come out of his letters to others and a few notes.
- 3.
Luca Pacioli (c. 1447–1517) was born in Borgo Santo Sepolcro in Tuscany and almost certainly received an abaco training from the painter Piero della Francesca. When he was around 20, Pacioli moved to Venice and worked for a wealthy merchant as tutor to his sons and possibly as his bookkeeper. In 1470 he moved to Rome, and at some point in the next five years, he became a Franciscan, which enabled him to work as an academic. In 1475 he moved to Perugia and started teaching mathematics privately, and then, between 1478 and 1480, at the city’s university.
Between 1481 and 1489, he was an itinerant math teacher until he returned to Santo Sepolcro, where della Francesca died in 1492. While at Santo Sepolcro, Pacioli would write his most important book, the abaco textbook, Summa de arithmetica, geometria, proportioni et proportionalita (‘Work on arithmetic, geometry and proportion’) that was published in Venice in 1494. The Summa was essentially a padded out version of Fibonacci’s Liber Abaci and is the first place that many of the ideas on which modern-day accounting is based were published. Concepts such as double-entry bookkeeping, the idea that a transaction is a debit in one book and a credit in another and the balance sheet appear in the Summa.
In 1497, Pacioli was invited to work in Milan by its Duke, Lodovico Sforza. There he met and collaborated with Leonardo da Vinci, who Sforza had also invited to Milan in 1482. Pacioli and da Vinci remained in Milan until 1506, when a French army captured the city and expelled Sforza and his court. Pacioli, approaching his 60s, seems to have returned to Santo Sepolcro where he died in 1517.
- 4.
Girolamo Cardano (1501–1576) was born the illegitimate but acknowledged son of Fazio Cardano who was a lawyer and the geometry lecturer at the University of Pavia, who probably knew da Vinci and Pacioli. Girolamo grew up during the devastating Italian Wars. In 1526 he became a Doctor of Medicine. The following year he moved to a small village where he worked as a doctor. In 1529 he applied to become a member of the College of Physicians of Milan, but was rejected, ostensibly, on the grounds of his illegitimacy. However, since illegitimacy was common at the time, it is unlikely that this was the real reason for his rejection. A more likely cause is that while in Padua, rumours about his sexuality spread and he developed a reputation as a drinker and gambler, playing a form of chess where the stakes were doubled as the game progressed, as in backgammon. Although he was able to practise medicine, on account of his degree from Padua, entering the College of Physicians would have given him the opportunity to treat wealthier patients. In 1531 he married, possibly to counter rumours of homosexuality, and a year later he applied again to the College, and was again rejected. Times were hard, and in 1534 he, his wife and child found themselves in the poorhouse.
Since he was unable to support himself as a physician, a friend rescued Cardano by securing for him the lectureship in geometry at the University of Milan. The next decade or so, up until around 1547, was Cardano’s most mathematically productive time. He established his reputation as a mathematician in 1539 by publishing Practica Arithmetica et Mensurandi Singularis (‘Practical arithmetic and simple measurement’). This book was essentially Pacioli’s, and has left Cardano with the reputation of a plagiarist. However, in an age when the primary objective of a university mathematician was to study astrology and astronomy, Cardano can be seen as bringing the practical mathematics of the abaco into the academic sphere.
Cardano’s wife died in 1546, but the years up to 1560 were Cardano’s best. On the basis of his mathematical exploits, he had fame and respect, which in turn brought him money . He was asked to cross Europe and treat John Hamilton, the Scottish Archbishop of St Andrews, and to provide a horoscope for the young English king, Edward VI, predicting a long life, the young man died the following year. One of his most enduring discoveries is the Cardan shaft, which is used in machines to transfer rotary motion at variable angles, and is still used in cars today and Cardano is regarded as being the first person to consider the meaning of \( \sqrt{-1} \), introducing imaginary numbers to mathematics.
In 1560, his life collapsed. His son, Giovanni, had married a Milanese prostitute in 1557 and when she did not give up her job, poisoned her. Despite his father’s best efforts, Giovanni was executed. By now Cardano was Professor of Mathematics at the University of Pavia, but following his son’s execution, his lectures became incoherent and he returned to the lifestyle he had had while studying at Padua; drinking, gambling and entertaining young men. He was forced to leave Pavia in 1562 but a former student, who ironically would be poisoned by his sister with the same poison that Cardano’s son had used, found him a job at Italy’s most prestigious university, Bologna. However, his teaching was getting worse and he was dismissed in 1570. This was not the bottom of his decline; he was imprisoned for impiety on account of having cast Christ’s horoscope in 1539, with his remaining son, Aldo, being involved in the prosecution. Casting Jesus’ horoscope was not an uncommon act, and the charge probably reflected family tensions. He was released after a short time and moved to Rome, where he died in 1576.
- 5.
Christiaan Huygens (1629–1695) was born into a wealthy and influential Dutch family, and between 1645 and 1649 he studied law and mathematics at the Universities of Leiden and Breda. His teacher, Frans van Schooten, had met Descartes and introduced Huygens to the Cartesian method and mathematics. In the second half of 1655, Huygens visited Paris and was told about the Problem of Points, but apparently not of its solution. He returned to the Netherlands, and inspired by these new ideas, wrote Van Rekeningh which would appear in van Schooten’s Exercitatonium Mathematicarum, essentially a university textbook, as De Ratiociniis in Ludo Aleae in 1657. Huygens work on probability was supplemented by work on physics and in 1662 he was elected a Fellow of the Royal Society of London and in 1666 he moved to Paris and became a member of the Académie Royal.
- 6.
Corpro relates to a body (corporation, corpus), while supra relates to being ‘above’ or ‘beyond’. So the corpro/supracorpo structures relate to tranching in modern-day CDOs.
- 7.
St Albert the Great (Albertus Magnus, c. 1200–1280) was born in Bavaria and went to the University of Padua, where he was introduced to the works of Aristotle, and in his early 20s he became a Dominican and moved to Bologna. In 1245 Albert moved to Paris and received a doctorate in theology and where he became interested in the relationship between God and the physical world. As natural philosophers, Albert and his fellow scholars, believed that natural phenomena had natural causes, but as Christians they believed God had some control, whether in design or in manifestation, of the same natural phenomena. Albert was able to reconcile science with religion, saying ‘Natural science does not consist in ratifying what others have said, but in seeking the causes of phenomena’. Albert became interested in economic questions when he started writing a commentary of Aristotle’s Nicomachean Ethics shortly after it was translated into Latin in 1250 by Robert Grosseteste.
- 8.
Pierre Jean Olivi (c. 1248–1298) was born near Béziers in Languedoc. Olivi entered the Franciscan order when he was 12 and was sent to Paris to study theology in 1267, and although he spent four years at the University, he did not graduate with a master’s degree. When he left Paris, he appears to have started working on a theological text that took him over 20 years to complete and addressed a range of questions, including the nature of free will. During this time he travelled widely in southern France and Italy and came into conflict with the Church hierarchy. Franciscans had an oath of poverty, and within a couple of generations of the founding of the order this oath began to be re-interpreted. Some Franciscans took the view that they kept to the oath if they did not own anything, others believed that this was a loophole; the oath required that a Franciscan should limit their use of goods. Olivi became a leader of this ‘rigorist’ or ‘spiritual’ wing of the Order. In 1282 he was accused of heresy and his writings destroyed, though he successfully defended himself in 1287 and was able to carry on teaching until his death in 1298. However, his tomb quickly attracted pilgrims and the Church, faced with a growing cult banned his writings in 1299, destroyed his tomb in 1312, and finally, when the Holy Roman Emperor, Louis the Bavarian, used some of Olivi’s arguments to attack the Papacy, he was condemned, again, as a heretic and all his works were obliterated.
- 9.
After October 1987, financial analysts started to observe that the market prices of options did not reflect the model prices in a consistent way, manifested in the ‘volatility smile’ or the ‘volatility skew’. The analysts ‘reverse engineered’ the BSM equation and extracted the ‘implied volatility’ by using observed option prices, quoted in the markets, as an input to the model and extracting the key parameter, the ‘implied volatility’. In theory, the implied volatility should be independent of the options’ strikes, but it was not. The deviations were caused by traders anticipating greater price moves than predicted by the asset price model of BSM.
- 10.
In A, the hedge funds had an insurable interest; B, ratings agencies had a bigger influence on pricing mortgage default than CDS; C, banks win and lose with CDS—that’s the point of market-making; D, disassociating pay-out from loss does create moral hazard if there is an insured interest, standardising pay-out enables price discovery.
- 11.
The Gaussian copula is used to describe the dependence of one random variable on another and had been identified in relation to the problem of estimating the lifespan of someone after their partner had died. Banks used it to model their portfolios as being made up of infinitely many, infinitesimally small identical loans. Each loan had an intrinsic probability of default but this was modified by the ‘correlation’ between defaults, ‘rho’, which represented the dependence of one loan defaulting on the default of another loan.
- 12.
Ducaton shares were reported in José de la Vega’s Confusion de Confusiones (1688). They had a nominal value of one tenth a Dutch East India Company (VOC) share, but there was no expectation that holding ten ducatons would entitle someone to a VOC share. Ducaton shares appeared because it was impossible for the public to participate in speculation on VOC shares, which were held exclusively by the Dutch elite and their trading incurred substantial transaction costs . Ducatons were a means through which the public could challenge the VOC owners’ assessment of the value of the firm and undermined the VOC shareholders’ assessment of their share valuations.
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Johnson, T. (2018). Teaching Reciprocity as the Foundation of Financial Economics. In: Feraboli, O., Morelli, C. (eds) Post-Crash Economics. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-65855-1_9
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